Shear-flow fields

Consider an arbitrarily oriented surface element in a homogeneous simple shear flow field vx = yyxy [Fig. 7.1a]. The surface element at time to is confined between two position vectors pi and p2. The area of the surface element is... 

Fig. E7.1a Surface element confined between position vectors pj and p2 in a simple shear flow field vx — y y. (a) At time tg. (b) At a later time t. ...

A single droplet of liquid deformed into a spheroid in a homogeneous shear flow field. 

Tubular flow reactors (TFR) deviate from the idealized PFR, since the applied pressure drop creates with viscous fluids a laminar shear flow field. As discussed in Section 7.1, shear flow leads to mixing. This is shown schematically in Fig. 11.9(a) and 11.9(b). In the former, we show laminar distributive mixing whereby a thin disk of a miscible reactive component is deformed and distributed (somewhat) over the volume whereas, in the latter we show laminar dispersive mixing whereby a thin disk of immiscible fluid, subsequent to being deformed and stretched, breaks up into droplets. In either case, diffusion mixing is superimposed on convective distributive mixing. Figure 11.9(c) shows schematically the... 

Based on thermodynamic considerations, criteria for the existence of domains in the melt in simple shear fields are developed. Above a critical shear stress, experimental data for the investigated block copolymers form a master curve when reduced viscosity is plotted against reduced shear rate. Furthermore the zero shear viscosity corresponding to data above a critical shear stress follow the WLF equation for temperatures in a range Tg + 100°C. This temperature dependence is characteristic of homopolymers. The experimental evidence indicates that domains exist in the melt below a critical value of shear stress. Above a critical shear stress the last traces of the domains are destroyed and a melt where the single polymer molecules constitute the flow units is formed in simple shear flow fields. 

Solid particles dispersed in a liquid have a two-particle collision frequency due to Brownian movement or to a shear flow field in the suspension after stirring, mixing, pumping or otherwise. 

Traditionally, shear viscosity measurements are used to rheologically characterize fluids. Eigure 6.1 shows the principle for shear viscosity measurement this figure shows a steady shear flow field between two parallel plates, one of which is moving with a velocity v. The measured quantities are the velocity of the top plate, the separation gap d, and the force in the direction of shear experienced by the stationary plate. Equation 6.1 is used to calculate the shear viscosity of the fluid, and the shear rate is calculated y = v/d (velocity/distance between the two parallel plates). Shear rate is also called velocity gradient. We can see that this shear rate or velocity gradient is constant. In this case, the displacement (strain) is... 

J. A. Schonberg and E. J. Hinch, Inertial migration of a sphere in Poiseuille flow, J. Fluid Mech. 203, 517-524 (1989) E. S. Asmolov, The inertial lift on a small particle in a weak-shear parabolic flow, Phys. of Fluids 14, 15-28 (2002) P. Cherukat, J. B. McLaughlin, and D. S. Dandy, A computational study of the inertial lift on a sphere in a linear shear flow field, Int. J. of Multiphase Flow, 25, 15-33 (1999). 

The microrheology makes it possible to expect that (i) The drop size is influenced by the following variables viscosity and elasticity ratios, dynamic interfacial tension coefficient, critical capillarity number, composition, flow field type, and flow field intensity (ii) In Newtonian liquid systems subjected to a simple shear field, the drop breaks the easiest when the viscosity ratio falls within the range 0.3 < A- < 1.5, while drops having A- > 3.8 can not be broken in shear (iii) The droplet breakup is easier in elongational flow fields than in shear flow fields the relative efficiency of the elongational field dramatically increases for large values of A, > 1 (iv) Drop deformation and breakup in viscoelastic systems seems to be more difficult than that observed for Newtonian systems (v) When the concentration of the minor phase exceeds a critical value, ( ) >( ) = 0.005, the effect of coalescence must be taken into account (vi) Even when the theoretical predictions of droplet deformation and breakup... 

Electron diffraction studies of Li-doped PPP have been carried out by Stamm et al. [180]. They use oriented films of Kovacic-type PPP prepared in a shear flow field, and Li vapour. The doped PPP shows some amorphous components, but the position of the reflections is essentially unchanged. It is therefore proposed that the lithium ions simply fill the vacancies between the chains in the crystal, along both the a and b cell edges (but with an occupancy of 0.5). The cell becomes orthorhombic upon doping. 

Shear flow fields are a remarkably effective means of orienting copol5mier microdomains. Large amplitude oscillatory shear (LAOS) and extrusion are common methods used to align... 

Chcmikat P, McLaughlin JB (1994) The inertial lift on a rigid sphere in a linear shear-flow field near a flat wall. J Hnid Mech 263 1-18... 

Fig. 7.2 Illustration of the definition to the Newtonian fluid in the shear flow field. / is the shear force, and A is the shear area...

Extensional flow is important for the dispersion process. As the microrheology indicates, the minimum of the k versus X curve is very narrow for the shear flow, but very broad (and lower) for the extensional flow (see Fig. 7.45). This suggests that it should be much easier to disperse fluids in extensional than in shear flow fields. 

Here, three important phenomena to control the phase separation morphology were explained the phase diagram and the phase separation in a shear flow field, reaction-induced phase separation, and reactive blending. 

Mohr et al. [20] have introduced the concept of a striation thickness (f) for shear flows, which was based on Spencer and Wiley s work [18]. The value of r is calculated by using the following equation when the melt viscosity difference between two components exists in the shear flow field ... 

A shear flow field can facilitate the formation of a shish-kebab stmcture, which consists of shish (extended chains that align along the flow direction) and kebabs (transversely grown lamellae perpendicular to shish). The skin layers are usually dominated by shish-kebabs.20... 

Two separate theoretical analyses of polymer molecular behavior in solution have been carried out by Bueche (12) and Graessley (13). Both analyses predict a pseudoplastic rheology for polymer solutions in simple shear. However, Bueche s explanation is based on the response of independent polymer molecules to a shearing flow field and Graessley s explanation is based on the response of entangled molecules when placed in a shearing flow field. 

When two viscous liquids are mixed, the interfacial area increases and the striation thickness decreases. Spencer and Wiley [201] have proposed to use the interfacial area as a quantitative measure of the goodness of mixing. Mohr et al. [189] used the striation thickness to describe the mixing process. If a surface element with arbitrary orientation is located in a simple shear flow field, the surface area A after a total shear strain of y can be demonstrated to be [201] ... 

Equation 7.479 indicates that the increase in interfacial area is directly proportional to the total shear strain and cosa. Thus, the total shear strain is an important variable in the description of the mixing process in a shear flow field. The initial orientation is also very important. If the initial surface is oriented parallel to the flow field (ttx = 90°), then the increase in interfacial area is zero. However, if the initial surface orientation is perpendicular to the flow field (a = 0), then the increase in interfacial area is maximum. At low strains, it can be seen from Eq. 7.477 that the interfacial area can increase or decrease with strain, depending on the initial orientation. 

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